Note that we write $F(2\Omega)$ to mean "apply $F$ after doubling the input".

The generators $X, X(2\Omega),f_2(2\Omega), K(2\Omega)$ are algebraically independent.
Let $B={\Bbb C}[X, X(2\Omega),f_2(2\Omega), K(2\Omega)]$. The module of modular forms with character of even weights only is then give by
$$
\oplus_{k=0}^\infty M_{2k}(\Gamma_0(4),\psi_4) =
f_{11}(2\Omega)f_{1}(2\Omega)B + Y(2\Omega) B + f_{11}(2\Omega)f_{3}(2\Omega)B,
$$
where $f_1=\theta_{0000}^2$ and $f_3=(\theta_{0000}\theta_{0001}\theta_{0011})^2$.

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.